In this work, we have shown that the PRHM scenario within the SI invariance
approach is possible; and in a good agreement with all the ATLAS and
CMS measurements. To avoid the constraints from the total Higgs signal
strength modifier [1]}, one needs to consider
the radiative corrections to the singlet-doublet scalar mixing. This
leads to non-negligible values for the triple scalar couplings \(\lambda _{HHH}\) 
and \(\lambda _{HHS}\) , and makes the PRHM scenario sensible to recent
ATLAS measurement of \(pp\rightarrow S\rightarrow HH\)  [2]}.
We considered the SI-scotogenic model [3]} as
an illustrative example, where we have checked different experimental
constraints and given some predictions about (Z-associated) di-Higgs
production at (\(ILC500\) ) \(LHC14\) . This PRHM scenario looks interesting
since many physical observables are all triggered together by the radiative corrections effects. Therefore,
some other aspects within this approach should be heavily investigated such as the electroweak
phase transition (EWPT) strength, gravitational waves produced during
the EWPT in addition to the different collider signatures that are
relevant to the triple Higgs couplings.
